High-performance ecc decoder

ABSTRACT

Methods for Error Correction Code (ECC) decoding include producing syndromes from a set of bits, which represent data that has been encoded with the ECC. An Error Locator Polynomial (ELP) is generated based on the syndromes. At least some of the ELP roots are identified, and the errors indicated by these roots are corrected. Each syndrome may be produced by applying to the bits vector operations in a vector space. Each syndrome is produced by applying vector operations using a different basis of the vector space. The ELP may be evaluated on a given field element by operating on ELP coefficients using serial multipliers, wherein each serial multiplier performs a sequence of multiplication cycles and produces an interim result in each cycle. Responsively to detecting at least one interim result indicating that the given element is not an ELP root, the multiplication cycles are terminated before completion of the sequence.

CROSS-REFERENCE TO RELATED APPLICATIONS

This application is a continuation of U.S. application Ser. No.13/920,140, filed Jun. 18, 2013, which is a continuation of U.S. patentapplication Ser. No. 13/590,565, filed Aug. 21, 2012, which is acontinuation of U.S. patent application Ser. No. 12/419,304, filed Apr.7, 2009, which claims the benefit of U.S. Provisional Patent ApplicationNo. 61/043,734, filed Apr. 10, 2008, U.S. Provisional Patent ApplicationNo. 61/043,736, filed Apr. 10, 2008, U.S. Provisional Patent ApplicationNo. 61/061,685, filed Jun. 16, 2008, and U.S. Provisional PatentApplication No. 61/105,454, filed Oct. 15, 2008. The disclosures of allthese related applications are incorporated herein by reference.

FIELD OF THE INVENTION

The present invention relates generally to Error Correction Coding(ECC), and particularly to methods and systems for high-performance ECCdecoding.

BACKGROUND OF THE INVENTION

Error Correction Codes (ECC) are used in a variety of applications, suchas in various digital communication and data storage applications. SomeECC decoders apply a multistage process, which decodes ECC code words bycalculating syndromes of the code words, and using the syndromes togenerate Error Locator Polynomials (ELPs) whose roots indicate the errorlocations in the code words, finding the ELP roots and correcting theerrors. Some ECC types that are commonly decoded using such a processcomprise, for example, Bose-Chaudhuri-Hocquenghem (BCH) codes andReed-Solomon (RS) codes.

Various schemes for generating ELPs from syndromes are known in the art.Some well-known schemes comprise, for example, the Berlekamp-Masseyalgorithm, the Euclidean algorithm and the Peterson Gorenstein Zierleralgorithm. Examples of methods for determining ELPs are described, forexample, by Lin and Costello in “Error Control Coding Fundamentals,”Prentice Hall, second edition, 2004, chapter 6, pages 209-215 andchapter 7, pages 241-255, and by Blahut in “Algebraic Codes for DataTransmission,” Cambridge University Press, 2004, chapter 6, pages131-166 and chapter 7, pages 179-190 and 217-223, which are incorporatedherein by reference.

A method for finding ELP roots is described by Chien in “Cyclic DecodingProcedure for the Bose-Chaudhuri-Hocquenghem Codes,” IEEE Transactionson Information Theory, vol. IT-10, October, 1964, pages 357-363, whichis incorporated herein by reference. This method is commonly known asthe “Chien search.”

SUMMARY OF THE INVENTION

An embodiment of the present invention provides a method for decoding anError Correction Code (ECC), including:

using hardware-implemented logic, producing from a set of bits, whichrepresent data that has been encoded with the ECC, multiple syndromes byapplying to the bits vector operations in a vector space, wherein eachsyndrome is produced by applying the vector operations to the set ofbits using a respective, different basis of the vector space;

generating, based on the multiple syndromes, an Error Locator Polynomial(ELP) whose roots are indicative of locations of respective errors inthe set of bits; and

identifying at least some of the roots of the ELP and correcting theerrors indicated by the identified roots.

In some embodiments, producing the syndromes includes selecting eachbasis such that the vector operations used for producing the respectivesyndrome comprise a multiplication of a sparse matrix. In an embodiment,the syndromes are defined over a field having a primitive element, andselecting each basis includes defining a set of basis elements asrespective multiples of a given vector by different powers of theprimitive element of the field. In a disclosed embodiment, the fieldincludes a Galois field. In an embodiment, after producing thesyndromes, the syndromes are transferred to a common basis of the vectorspace.

There is additionally provided, in accordance with an embodiment of thepresent invention, a method for decoding an Error Correction Code (ECC),including:

accepting coefficients of an Error Locator Polynomial (ELP), which isdefined over a field and whose roots are indicative of locations ofrespective errors in a set of bits, which represent data that has beenencoded with the ECC;

evaluating the ELP on a given element of the field by operating on thecoefficients using respective hardware-implemented serial multipliers,such that each serial multiplier performs a sequence of multiplicationcycles and produces an interim result in each cycle;

responsively to detecting, during the sequence of the multiplicationcycles, at least one interim result indicating that the given element isnot one of the roots of the ELP, terminating the multiplication cyclesbefore completion of the sequence; and

when the interim results indicate that the given element is a root ofthe ELP, correcting at least one error indicated by the given element.

In some embodiment, evaluating the ELP includes applying the ELPconcurrently to multiple elements of the field using respective multiplesets of the serial multipliers. In a disclosed embodiment, terminatingthe multiplication cycles includes terminating the multiplication cyclesapplied to the multiple elements responsively to determining, based onthe interim results, that none of the multiple elements comprises an ELProot. In another embodiment, terminating the multiplication cyclesincludes terminating the multiplication cycles applied to one of themultiple elements irrespective of termination of the multiplicationcycles applied to the other elements.

In yet another embodiment, when a rank of the ELP does not exceed halfof a number of the serial multipliers in each of the sets of the serialmultipliers, applying the ELP includes dividing each of the sets of theserial multipliers into first and second subsets, and applying the ELPconcurrently to respective first and second elements of the field usingthe first and second subsets of the serial multipliers.

There is also provided, in accordance with an embodiment of the presentinvention, a method for decoding an Error Correction Code (ECC),including:

in an ECC decoder that includes multiple logic components that areclocked by a clock signal, accepting an Error Locator Polynomial (ELP),which has a given rank and is defined over a field, and whose roots areindicative of locations of respective errors in a set of bits, whichrepresent data that has been encoded with the ECC;

based on the rank of the ELP, selectively disabling the clock signal tosome of the logic components that are used for computing the roots ofthe ELP;

identifying the roots of the ELP using the logic components for whichthe clock signal has not been disabled; and

correcting the errors indicated by the identified roots.

In some embodiments, the logic components are arranged in a first numberof subsets, each of which is assigned to process a respective ELPcoefficient, the ELP includes a second number of ELP coefficients,smaller than the first number, and selectively disabling the clocksignal includes providing the clock signal only to the subsets that areassigned to process the second number of the coefficients. In adisclosed embodiment, a rate of the clock signal is modifiedresponsively to the rank of the ELP. In another embodiment, the methodincludes, upon identifying a root of the ELP, dividing the ELP by afactor that depends on the identified root to produce a lower-rank ELP,and continuing to identify the roots of the lower-rank ELP. In yetanother embodiment, the logic components include at least one componenttype selected from a group of types consisting of multipliers andregisters.

There is further provided, in accordance with an embodiment of thepresent invention, an Error Correction Code (ECC) decoder, including:

a syndrome calculation unit, which is coupled to produce from a set ofbits, which represent data that has been encoded with the ECC, multiplesyndromes by applying to the bits vector operations in a vector space,wherein each syndrome is produced by applying the vector operations tothe set of bits using a respective, different basis of the vector space;

an Error Locator Polynomial (ELP) computation unit, which is configuredto generate, based on the multiple syndromes, an ELP whose roots areindicative of locations of respective errors in the set of bits; and

a root search unit, which is coupled to identify at least some of theroots of the ELP so as to correct the errors indicated by the identifiedroots.

There is additionally provided, in accordance with an embodiment of thepresent invention, an Error Correction Code (ECC) decoder, including:

root search circuitry, which includes multiple serial multipliers and isconfigured to accept coefficients of an Error Locator Polynomial (ELP),which is defined over a field and whose roots are indicative oflocations of respective errors in a set of bits, which represent datathat has been encoded with the ECC, and to evaluate the ELP on a givenelement of the field by operating on the coefficients using respectiveones of the serial multipliers, such that each serial multiplierperforms a sequence of multiplication cycles and produces an interimresult in each cycle; and

control logic, which is configured to terminate the multiplicationcycles before completion of the sequence responsively to detecting,during the sequence of the multiplication cycles, at least one interimresult indicating that the given element is not one of the roots of theELP, and to correct at least one error indicated by the given elementwhen the interim results indicate that the given element is a root ofthe ELP.

There is further provided, in accordance with an embodiment of thepresent invention, an Error Correction Code (ECC) decoder, including:

root search circuitry, which includes multiple logic components that areclocked by a clock signal, and is coupled to accept an Error LocatorPolynomial (ELP), which has a given rank and is defined over a field,and whose roots are indicative of locations of respective errors in aset of bits, which represent data that has been encoded with the ECC,and to identify the roots of the ELP using the logic components so as tocorrect the errors indicated by the identified roots; and

a control unit, which is configured to selectively disable the clocksignal to some of the logic components based on the rank of the ELP, soas to cause the root search circuitry to identify the roots of the ELPusing only the logic components for which the clock signal has not beendisabled.

The present invention will be more fully understood from the followingdetailed description of the embodiments thereof, taken together with thedrawings in which:

BRIEF DESCRIPTION OF THE DRAWINGS

FIG. 1 is a block diagram that schematically illustrates a communicationsystem that employs Error Correction Coding (ECC), in accordance with anembodiment of the present invention;

FIG. 2 is a block diagram that schematically illustrates a data storagesystem that employs ECC, in accordance with an embodiment of the presentinvention;

FIG. 3 is a block diagram that schematically illustrates an ECC decoder,in accordance with an embodiment of the present invention;

FIG. 4 is a block diagram that schematically illustrates a syndromecalculation unit, in accordance with an embodiment of the presentinvention;

FIG. 5 is a flow chart that schematically illustrates a method forsyndrome calculation, in accordance with an embodiment of the presentinvention;

FIG. 6 is a block diagram that schematically illustrates an ELP rootcalculation unit, in accordance with an embodiment of the presentinvention;

FIG. 7 is a flow chart that schematically illustrates a method for ELProot calculation, in accordance with an embodiment of the presentinvention;

FIGS. 8 and 9 are block diagrams that schematically illustrate ELP rootcalculation units, in accordance with alternative embodiments of thepresent invention; and

FIGS. 10 and 11 are flow charts that schematically illustrate methodsfor locating ELP roots, in accordance with embodiments of the presentinvention.

DETAILED DESCRIPTION OF EMBODIMENTS Overview

Power consumption and hardware size are prime considerations in many ECCdecoding applications. For example, ECC decoders are commonly used insmall and low-cost communication, computing and storage devices, whichoperate on battery power. In these sorts of devices, it is important tominimize the power consumption and the physical size of the ECC decoder,in order to reduce the battery life, physical size and cost of thedevice.

Embodiments of the present invention provide improved methods andapparatus for ECC decoding. These techniques achieve considerablereduction in hardware size and power consumption in comparison withknown solutions. The disclosed techniques are suitable for various kindsof ECC that use syndromes and Error Locator Polynomials (ELPs), such asBose-Chaudhuri-Hocquenghem (BCH) codes and Reed-Solomon (RS) codes.

In a typical embodiment, an ECC decoder accepts input code words thatmay contain errors. The ECC decoder operates on each input code word toproduce multiple syndromes of the code word. Using the syndromes, theECC decoder generates an ELP, whose roots are indicative of locations oferrors in the code word. The ECC decoder then finds the ELP roots, andcorrects the errors indicated by the roots.

The computation of a given syndrome can be represented as amultiplication of a matrix by a vector in a certain vector space. Insome embodiments, the ECC decoder is designed so that the vectoroperations (e.g., multiplications) associated with the computation ofeach syndrome are performed using a different basis of the vector space.Each basis is selected such that the multiplied matrix is sparse, i.e.,has only a small number of non-zero elements. As a result, themultiplications can be implemented using smaller-size hardware thatconsumes less power.

Typically, the ELP is defined over a Galois Field (GF), and the ECCdecoder finds the ELP roots by evaluating the ELP on different fieldelements. In some embodiments, the ECC decoder evaluates the ELP byoperating a set of serial multipliers, each of which operates on arespective ELP coefficient. Each serial multiplier operates on therespective coefficient in a sequence of computation cycles, and producesan interim result in each cycle. The ECC decoder monitors the interimresults during the computation sequence. If, during the sequence, theECC decoder detects at least one interim result indicating that thecurrently-evaluated field element is not a root of the ELP, thecomputation sequence is terminated before its completion. Unlike ECCdecoders that use parallel multipliers in which the ELP is fullyevaluated for each field member, the disclosed technique terminates theevaluation process of a given field element as soon as the element isfound not to be an ELP root. As a result, the power consumption of theroot search process is reduced. Moreover, since serial multipliers aretypically smaller than comparable parallel multipliers, the hardwaresize of the disclosed configurations is relatively small.

The rank of a given ELP indicates the number of errors in thecorresponding code word. In the ECC decoder, the logic that searches forELP roots is typically dimensioned according to the maximum errorcorrection capability of the decoder, i.e., the maximum specified ELPrank. In many cases, however, the number of errors in a code word islower than the maximum number, and the actual ELP rank is thus lowerthan the maximum specified rank.

Therefore, in some embodiments, the ECC decoder reduces its powerconsumption by selectively deactivating clock signals provided to theroot search logic, based on the actual rank of the ELP. Using thistechnique, only the logic components (e.g., registers and multipliers)that actually participate in the root search of the actual ELP areprovided with clock signals. The ECC decoder may also modify the clockspeed based on the actual ELP rank, so as to further control thedecoder's power consumption. In some embodiments, when a certain ELProot is found during the search process, the decoder divides the ELP soas to factor out this root, and continues the search with the lower-rankELP. Using this technique, the decoder's power consumption decreaseswith time, as additional roots are found and the ELP rank is reduced.

Several efficient hardware configurations that implement the disclosedtechniques are described and discussed hereinbelow.

System Description

Embodiments of the present invention provide improved methods andsystems for decoding Error Correction Codes (ECC), such asBose-Chaudhuri-Hocquenghem (BCH) or Reed-Solomon (RS) codes. Thedisclosed techniques can be used in a wide variety of systems andapplications in which ECC is deployed, such as in various communicationand data storage systems. FIGS. 1 and 2 below illustrate two exampleapplications.

FIG. 1 is a block diagram that schematically illustrates a wirelesscommunication system 20 that employs error correction coding, inaccordance with an embodiment of the present invention. System 20comprises a transmitter 24, which transmits data to a receiver 28. Thetransmitter accepts input data, encodes the data with a certain ECC,modulates the encoded data in accordance with a certain modulationscheme, converts the modulated digital signal to an analog signal,up-converts the analog signal to a suitable Radio frequency (RF), andtransmits the RF signal toward the receiver using a transmit antenna 32.

In receiver 28, a receive antenna 36 receives the RF signal and providesit to a RF front end 40. The front end down-converts the RF signal tobaseband or to a suitable Intermediate Frequency (IF), and digitizes thesignal with a suitable Analog to Digital Converter (ADC—not shown in thefigure). The digitized signal carrying the ECC-encoded data isdemodulated by a modem 44, and the ECC is decoded by an ECC decoder 48.The performance of decoder 48 is enhanced by a processor 52, usingmethods that are described in detail below. By decoding the ECC, decoder48 reconstructs the data that was input to transmitter 24. Thereconstructed data is provided as the receiver output.

System 20 may comprise, for example, a cellular system, a satellitesystem, a point-to-point communication link, or any other suitablecommunication system that employs ECC. Although the example of FIG. 1refers to a wireless communication system, the techniques describedherein can be used with wire-line communication systems, such as cablecommunication systems, as well.

FIG. 2 is a block diagram that schematically illustrates a data storagesystem 60 that employs error correction coding, in accordance with analternative embodiment of the present invention. System 60 comprises amemory controller 64, which stores data in a memory device 68. Thememory device comprises an array 72 comprising multiple memory cells 76.Array 72 may comprise any suitable type of volatile or non-volatilememory, such as, for example, Random Access Memory (RAM) or Flashmemory. Alternatively, device 68 may comprise a magnetic storage devicesuch as a Hard Disk Drive (HDD), or any other suitable storage medium.System 60 can be used in various host systems and devices, such as incomputing devices, cellular phones or other communication terminals,removable memory modules (“disk-on-key” devices), Solid State Disks(SSD), digital cameras, music and other media players and/or any othersystem or device in which data is stored and retrieved.

Memory device 68 comprises a Read/Write (R/W) unit 80, which writes datavalues into memory cells 76 and reads data values from the memory cells.Memory controller 64 comprises an ECC unit 84, which encodes the datafor storage with a certain ECC, and decodes the ECC of data that isretrieved from the memory cells. The performance of unit 84 in decodingthe ECC is enhanced by a processor 88, using methods that are describedin detail below. The ECC used in systems 20 and 60 may comprise, forexample, a suitable BCH or RS code, as well as various other types ofECC.

Processors 52 and 88, ECC decoder 48 and ECC unit 84 can be implementedin software, in hardware or using a combination of hardware and softwareelements. In some embodiments, processors 52 and 88 comprisegeneral-purpose processors, which are programmed in software to carryout the functions described herein. The software may be downloaded tothe processors in electronic form, over a network, for example, or itmay, alternatively or additionally, be provided and/or stored ontangible media, such as magnetic, optical, or electronic memory.

The ECC decoding schemes described herein can be used in communicationsystems such as system 20, as well as in data storage systems such assystem 60. The description that follows applies to both communicationapplications and to storage applications, and refers generally to an ECCdecoder and a processor. Any reference to the ECC decoder applies todecoder 48 of system 20, as well as to the decoder functionality of unit84 in system 60. Any reference to the processor applies to processor 52of system 20, as well as to processor 88 in system 60. Alternatively,the methods described herein can be carried out by any suitable elementin any suitable system that involves ECC decoding.

FIG. 3 is a block diagram that schematically illustrates an ECC decoder90, in accordance with an embodiment of the present invention. ECCdecoder 90 accepts ECC code words, which may contain errors. In otherwords, the input code words may not always comprise valid code words ofthe ECC. The ECC decoder decodes the input code words while attemptingto correct these errors, so as to reconstruct the data conveyed in thecode words. In a typical implementation, each code word comprises on theorder of several hundred to several thousand bits, although any othersuitable code word size can be used.

Decoder 90 comprises a syndrome calculation unit 94, which calculates asyndrome for each input code word. The syndrome is typically defined asHy=S, wherein H denotes the parity check matrix of the ECC, y denotes aninput code word and S denotes a vector of T syndromes of code word y,denoted S₁ . . . S_(T). T denotes the maximum number of errors that theECC is able to correct per code word. When input code word y contains noerrors (i.e., when y is a valid code word), Hy=0.

When the ECC is defined over a certain finite Galois Field (GF) having aprimitive field element (also referred to as a field-generating element)α, the k^(th) syndrome S_(k) can typically be written as S_(k)=Σ_(i=0)^(n−1)b_(i)α^(ki) (or as S_(k)=Σ_(i=0) ^(n−1)b_(i)α^(n−1−ki) if the bitorder is reversed), wherein b_(i) denote the bits of the input codeword. For a BCH code, coefficients b_(i) are elements of the fieldGF(p), and each syndrome S_(k) is an element of the field GF(p^(m)). Fora Reed-Solomon code, both coefficients b_(i) and the syndromes S_(k) areelements of the field GF(p^(m)). The description that follows refersmainly to codes defined over GF(2^(m)), although the methods and systemsdescribed herein are applicable to codes defined over any other suitablefield. In a typical implementation, m=4 (2^(m)=16), although any othersuitable value of m can also be used. Unit 94 typically computes andoutputs a set of T syndromes for each input code word.

(The description given herein refers mainly to BCH and RS codes,although it is applicable to various other codes that are defined bymultiples of a given polynomial. In general, the syndromes are producedby applying the roots of this polynomial to the received code word.)

The syndromes output by unit 94 are processed by an Error LocatorPolynomial (ELP) computation unit 98. For a given set of T syndromescorresponding to a given code word, unit 98 determines an ELP definedover GF(2^(m)) whose roots are indicative of the error locations in thegiven code word. The ELP can be written as ELP(x)=1+α₁x+α₂x²+ . . .+α_(j)x^(j), wherein j denotes the rank of the ELP, j≦T. Unit 98 mayapply any suitable method in order to compute the ELP for a given codeword, such as, for example, the Berlekamp-Massey method, the Euclideanmethod or the Peterson Gorenstein Zierler method, cited above.

Unit 98 provides the ELPs to an ELP root search unit 102. Unit 102determines the roots of each ELP, i.e., the elements x of GF(2^(m)) forwhich ELP(x)=0. For a given code word, the ELP roots are indicative ofthe locations of the errors within the code word. ECC decoder 90 thencorrects the errors at the identified locations. For a BCH code, the ELProots identify the erroneous bits in the code word, and the decodercorrects the errors by reversing the values of the identified bits. In aRS code, on the other hand, the ELP roots indicate the erroneous symbolsin the code words. In this case, decoder 90 determines the error valuesin addition to the error locations in order to correct the errors. Thecorrection functionality can be carried out either by unit 102 or byother circuitry (not shown in the figure) in decoder 90.

Typically, units 94, 98 and 102 are implemented in hardware, such asusing one or more Application-Specific Integrated Circuits (ASICs),Field-Programmable gate Arrays (FPGAs) and/or discrete components. Someor all of the decoder functions may alternatively be implemented insoftware, or using a combination of software and hardware elements. Anefficient configuration for implementing unit 94 is described in FIGS. 4and 5 below. Several efficient schemes for locating ELP roots aredescribed in FIGS. 6-11 below.

Efficient Syndrome Calculation

As noted above, a syndrome S_(k) can be expressed as S_(k)=Σ_(i=0)^(n−1)b_(i)α^(ki), wherein b_(i) denote the bits of the input code wordand a denotes α primitive element of the Galois field. The syndrome canbe calculated bit-by-bit by calculating S_(k) ^(i+1)=S_(k)^(i)α^(k)+b_(i). A more efficient calculation processes a group of rbits at a time by calculating

$\begin{matrix}{S_{k}^{i + r} = {{S_{k}^{i}\alpha^{kr}} + {\sum\limits_{l = 0}^{r - 1}\; {b_{i + r - 1 - l}\alpha^{rl}}}}} & \lbrack 1\rbrack\end{matrix}$

FIG. 4 is a block diagram that schematically illustrates the internalstructure of syndrome calculation unit 94, in accordance with anembodiment of the present invention. In the present example, unit 94comprises multiple syndrome calculation modules 106A, 106B . . . . Thesyndrome calculation modules operate in parallel on a given code word,such that each module calculates a respective syndrome of the code word.

In each clock cycle, each syndrome calculation module accepts r bitsfrom the code word, and carries out the calculation of Equation [1]above. Each syndrome calculation module in unit 94 comprises a register110, a Galois field multiplier 114, an adder 118 and logic 122. Considerthe k^(th) syndrome calculation module, which calculates syndrome S_(k).In a given iteration, register 110 of this syndrome calculation moduleholds the previous value of the syndrome (S_(k) ^(i)), and multipliesthis value by α^(kr) to produce the first term on the right-hand-side ofEquation [1]. Logic 122 calculates the second term (Σ_(l=0)^(r−1)b_(i+r−1−l)α^(rl)). Adder 118 adds the two terms, and stores theresult (S_(k) ^(i+r)) back in register 110. At the end of the iterativeprocess, register 110 holds the value of syndrome S_(k).

The task of multiplying S_(k) ^(i) by α^(kr) can be represented as a setof vector operations that implement matrix multiplication. In thisrepresentation, the set of m powers of the field-generating element αform a basis that spans GF(2^(m)). Thus, any element of the GF(2^(m))field can be represented as Σ_(i=0) ^(m−1)d_(i)α^(i), wherein d_(i)comprise binary bits, i.e., as an m-tuple of binary bits. In particular,since syndrome S_(k) is an element in field GF(2^(m)), it can berepresented as such an m-tuple. The task of multiplying a Galois fieldelement d, which is represented by the coefficients d₀ . . . d_(m−1), bya constant k, comprises a linear operation on the coefficients d₀ . . .d_(m−1). This task can therefore be carried out by a matrixmultiplication in GF(2), of the form c=k·d, such that c_(i)=Σ_(j=0)^(m−1)k_(ij)d^(i).

In unit 94, the above-mentioned matrix multiplication is carried out bymultiplier 114 in each syndrome calculation module. The multipliertypically comprises digital logic circuitry (e.g., multiple XOR gates orother logic), which carries out this multiplication. The hardwarecomplexity and power consumption of this logic circuitry grow with thenumber of non-zero elements of the matrix representing α^(kr).

In some embodiments, each syndrome calculation module reduces thecomplexity and power consumption of the syndrome calculation process bytransforming the calculation to a different vector space in which thematrix representing α^(kr) is sparse, i.e., contains a small number ofnon-zero elements. When the matrix representing α^(kr) is sparse, thetask of multiplying S_(k) ^(i) by α^(kr) comprises a relatively smallnumber of logic computations. As a result, the hardware (e.g., logicgates) carrying out this task in multipliers 114 can be reduced in size.The power consumption of the decoder is thus reduced, as well.

Transformation of the matrix to the desired vector space is performed byrepresenting the matrix by a different basis. One possible way totransform a general matrix A into a sparse representation is to selectan arbitrary vector y, and construct a basis of the form {y, Ay, . . . ,A_(i)y, . . . }. In other words, each basis element comprises vector y,multiplied by a different power of matrix A. The representation ofmatrix A using this basis is a matrix in which (1) one row (or column)has non-zero elements, (2) the elements in one of the off-diagonals areequal to unity, and (3) all other elements are zero. This scheme can begeneralized in a straightforward manner to rectangular matrices and tomatrices that do not fully span the vector space.

Using the above-described scheme, matrix α^(kr) (which is used by thek^(th) syndrome calculation module in calculating S_(k) ^(i)) istransformed to a basis whose elements comprise an arbitrary vector ymultiplied by different powers of α^(kr). For implementation reasons,vector y is typically selected as 1, although any other vector can alsobe selected. When y=1, the new basis has the form {1, α^(kr), α^(2kr),α^(3kr), . . . , α^((m−1)kr)}. (Typically, each syndrome is computedusing a different basis.)

The resulting matrix, after basis transformation, is an m-by-m binarymatrix having at most 2m−1 non-zero elements. The original matrix beforebasis transformation is typically balanced, i.e., has approximately m²/2non-zero elements. Thus, the basis transformation described abovereduces the hardware size and power consumption of multipliers 114 by afactor of ˜m/4. Moreover, before basis transformation the hardware sizeand power consumption are on the order of O(m²), and the basistransformation reduces them to an order of O(m).

When carrying the basis transformation described above, Equation [1]above takes becomes:

$\begin{matrix}{\left( S_{k}^{i + r} \right) = {{\begin{pmatrix}\left\lbrack \propto^{kr} \right\rbrack & \ldots & \left\lbrack \propto^{{kr} + m - 1} \right\rbrack\end{pmatrix}\left( S_{k}^{i} \right)} + {\begin{pmatrix}\lbrack 1\rbrack & \left\lbrack \propto^{k} \right\rbrack & \ldots\end{pmatrix}\begin{pmatrix}b_{i} \\b_{i + 1} \\\vdots\end{pmatrix}}}} & \lbrack 2\rbrack\end{matrix}$

wherein [∝^(k)] is a column vector of the coefficients of ∝^(k) in thestandard basis (i.e., before transformation).

The transformation matrix R, which transforms the new basis to theoriginal basis, comprises the columns of the new basis expressed usingthe original basis, i.e.:

$\begin{matrix}{R = \left( {{\lbrack 1\rbrack \left\lbrack \propto^{{kr} + 1} \right\rbrack}{\ldots \left\lbrack \propto^{{kr}{({m - 1})}} \right\rbrack}} \right)} & \lbrack 3\rbrack\end{matrix}$

The basis transformation is thus given by

{tilde over (S)} _(k) ^(i+r) =AS _(k) ^(i) +B{right arrow over (b)}  [4]

wherein A and B are given by:

$\begin{matrix}{A = {R^{- 1}\left( {{\left\lbrack \propto^{kr} \right\rbrack \left\lbrack \propto^{{kr} + 1} \right\rbrack}{\ldots \left\lbrack \propto^{{{kr} + m - 1})} \right\rbrack}} \right)}} & \lbrack 5\rbrack \\{B = {R^{- 1}\left( {{\lbrack 1\rbrack \left\lbrack \propto^{k} \right\rbrack}\ldots} \right)}} & \lbrack 6\rbrack\end{matrix}$

The transformation back from the new basis to the original basis isgiven by

S _(k) ^(i) =R{tilde over (S)} _(k) ^(i)  [7]

In summary, multiplier 114 of the k^(th) syndrome calculation module inunit 94 comprises circuitry, which multiplies S_(k) ^(i) by α^(kr) usingthe above-mentioned basis transformation that is applicable for thek^(th) syndrome S_(k). The multiplier may also comprise circuitry thattransforms the computed syndrome back to the original basis, or to anyother basis in which different syndromes can compared and furtherprocessed. For example, transformation back to the original basis can beperformed before providing the syndromes to ELP computation unit 98.Alternatively, unit 98 may consider the basis in which each syndrome isrepresented when computing the ELP. In the latter implementation,transformation to the original basis can be omitted.

Although the description above refers to BCH codes, the basistransformation scheme can be adapted to other cyclic block codes thatuse syndromes, such as RS codes. The above-mentioned process can beapplied for any desired value of r, i.e., for iterative processes thatcompute the syndromes based on one or more bits per iteration. Theabove-mentioned process can be applied in hardware, in software or usinga combination of hardware and software elements.

FIG. 5 is a flow chart that schematically illustrates a syndromecalculation method, carried out by unit 94, in accordance with anembodiment of the present invention. The method begins with syndromecalculation unit 94 accepting a code word, at an input step 130. Eachsyndrome calculation module (106A, 106B, . . . ) in unit 94 computes arespective syndrome of the input code word, at a syndrome computationstep 138. Each syndrome computation module computes the syndrome usingthe transformed basis that is applicable to this syndrome, as describedabove. Typically, each syndrome calculation module computes the syndromein an iterative process, which processes r bits of the code word in eachiteration (r≧1). In some embodiments, although not necessarily, thesyndrome calculation modules transform the syndromes back to theoriginal basis, at a backward transformation step 142. Unit 94 providesthe computed syndromes to ELP root search unit 102, at an output step146.

Efficient ELP Root Searching

As noted above, ELP root search unit 102 receives a set of ELPcoefficients for each input code word from ELP computation unit 98. Unit102 processes the ELP coefficients so as to identify the roots of theELP. As long as the number of errors in the code word does not exceed T,the ELP roots are indicative of the locations of errors in the inputcode word, and therefore locating the ELP roots enables correction ofthe errors. In the description that follows, the ELP is given byELP(x)=1+α₁x+α₂x²+ . . . +α_(j)x^(j), wherein coefficients a₁ . . .a_(j) are provided by unit 98. Unit 102 searches the GF(2^(m)) field inan attempt to find the field elements for which ELP(x)=0.

Evaluating ELP(x) for various field elements typically involvesmultiplying the ELP coefficients by elements of the GF(2^(m)) field, andmore specifically multiplying the ELP coefficients by powers of thefield-generating element α. Each such multiplication multiplies acertain field element by a certain ELP coefficient. Some known methodsperform these multiplications using parallel Galois Field (GF)multipliers, such that each multiplier multiplies a certain fieldelement by a certain ELP coefficient in a single clock cycle.

Embodiments of the present invention that are described below provideimproved root search configurations, which perform the above-mentionedmultiplications using serial GF multipliers. In the present context, theterm “serial multiplier” means any multiplier that multiplies a certainfield element by a certain ELP coefficient in multiple clock cycles, andoutputs one or more bits of the final product (typically one bit) ineach clock cycle. Examples of serial multipliers are described on pages265-267 of Blahut's “Algebraic Codes for Data Transmission,” CambridgeUniversity Press, 2004, which is incorporated herein by reference.Alternatively, however, any other suitable serial multiplierconfiguration can also be used.

The objective of the root search process is to find field elements forwhich ELP(x)=0. Therefore, even a single non-zero bit produced by theserial multiplier indicates that the multiplied field element is not aroot of the ELP. Thus, when a given serial multiplier outputs a bit ofthe product that is non-zero, the multiplication process can beterminated before its completion. This scheme reduces the number ofmultiplications performed in the root search process, since most of themultiplication operations, which multiply field elements that are notELP roots, are terminated before completion. As a result, the powerconsumption of the root search process is reduced considerably. Thehardware size (e.g., gate count) of a serial multiplier is alsoconsiderably smaller than that of a comparable parallel multiplier.

FIG. 6 is a block diagram that schematically illustrates an ELP rootcalculation unit 150, in accordance with an embodiment of the presentinvention. The configuration of unit 150 can be used to implement unit102 in FIG. 3 above. Unit 150 comprises T+1 coefficient multiplicationmodules 154. The k^(th) module 154 evaluates the k^(th) term of ELP(x),i.e., a_(k)x^(k), k=0 . . . T−1.

Each module 154 comprises parallel multipliers 158 and 162, a register166 and a serial multiplier 170. In the k^(th) module 154, parallelmultiplier 158 multiplies the content of register 166 by α^(k). Parallelmultiplier 162 multiplies the content of register 166 by α, and istherefore relatively simple and inexpensive. The two parallelmultipliers are operated only once per each evaluated field element, aswill be shown below.

In order to evaluate a_(k)x^(k) in the k^(th) module 154, register 166is initialized with the ELP coefficient a_(k). Then, a sequence of mcycles is performed, in which serial multiplier 170 multiplies thecontent of register 166 by α^(k). The m cycles are performedconcurrently in the T+1 modules 154. At the end of each cycle, eachserial multiplier 170 produces one bit of its final product. This bit isreferred to as an interim result.

In each cycle, a XOR unit 174 computes the Exclusive OR (XOR) of the T+1interim results produced by modules 154. Thus, if unit 174 outputs a “1”at any time during the m-cycle multiplication process, unit 150 mayconclude that the currently-evaluated field element is not an ELP root.In such a case, the multiplication process can be terminated, and unit150 can proceed to evaluate the next field element.

FIG. 7 is a flow chart that schematically illustrates a method for ELProot calculation, carried out by unit 150, in accordance with anembodiment of the present invention. Unit 150 scans the GF fieldelements (1, α, α², α³, . . . ) sequentially, in order to identify whichof the field elements are roots of the ELP.

The method begins with unit 150 initializing registers 166 with the ELPcoefficients, at an initialization step 180. Unit 150 now carries out asequence of m cycles, in which serial multipliers 166 multiply theirrespective ELP coefficients by the appropriate powers of thecurrently-evaluated field element.

Unit 150 operates the serial multipliers for a single cycle, at a cycleoperation step 184. At the end of this cycle, each serial multiplierproduces a respective interim result, i.e., one bit of the finalproduct. Unit 174 calculates a XOR of the T+1 interim results. Unit 174checks whether XOR result after the current cycle is non-zero, at anon-zero checking step 188. If the XOR operation produces a non-zerooutput, unit 174 terminates the m-cycle sequence and proceeds toevaluate the next field element, at a next candidate step 192. Whenproceeding to the next field element, parallel processors 158 and 162 ofmodules 154 are operated to calculate the next power of α. Unit 174 thusfunctions as control logic, which monitors the interim results of theserial multipliers and terminates the multiplication sequence whenappropriate.

If, on the other hand, the output of unit 174 is zero, the m-cyclemultiplication sequence continues. Unit 150 checks whether all m cycleshave been completed. If not, the method loops back to step 184 above inorder to proceed to the next cycle in the sequence. If all m cycles havebeen completed, and the output of unit 174 has been zero during theentire sequence, unit 150 identifies the currently-evaluated fieldelement as a root of the ELP, at a root identification step 200. Unit150 may correct the error at the location indicated by the identifiedroot, or report the identified root or error location to decoder 90. Themethod then loops back to step 192 above, in which unit 150 proceeds toevaluate the next field element. This process of FIG. 7 typicallycontinues until all field elements have been evaluated or until T rootshave been found.

FIG. 8 is a block diagram that schematically illustrates an ELP rootcalculation unit 204, in accordance with an alternative embodiment ofthe present invention. In this configuration, the ELP root calculationunit evaluates multiple GF field elements concurrently. Unit 204comprises T+1 coefficient multiplication modules 208. In thisconfiguration, however, each module 208 comprises d serial multipliers170. Each row of serial multipliers 170 in FIG. 204 evaluates arespective field element. The outputs of the serial multipliers in agiven row are connected to a respective XOR unit 212, which calculatesthe XOR of the T+1 interim results produced by the serial multipliers.

In the configuration of FIG. 8, a set of d field elements are evaluatedconcurrently in each m-cycle multiplication sequence of the serialmultipliers. Parallel multipliers 158 and 162 are operated only once pereach set of d field elements. Thus, the overhead of operating theparallel multipliers is divided among d field elements, instead of asingle field element as in FIG. 6 above. Thus, the configuration of FIG.8 provides improved power consumption and latency in comparison with theconfiguration of FIG. 6 above.

At any time during the m-cycle multiplication sequence, if a certain XORunit 212 produces a non-zero output, unit 204 may conclude that thefield element evaluated by the respective row of serial multipliers isnot an ELP root. When a given XOR unit 212 produces a non-zero output,unit 204 may terminate the operation of the respective row of serialmultipliers in order to reduce power consumption. Alternatively, unit204 may wait for a situation in which all XOR units produce non-zerooutputs, and then terminate the entire m-cycle sequence and proceed tothe next set of d field elements.

Thus, units 212 function collectively as control logic, which monitorsthe interim results produced by the serial multipliers and terminatesone or more of the multiplication sequences as appropriate.

The configurations of FIGS. 6 and 8 above are example configurations,which are chosen for the sake of conceptual clarity. Any other suitableconfiguration can also be used. For example, the field elements can bescanned in any desired order, such as by using other primitive fieldelements, or by multiplying by α⁻¹ instead of by α. As another example,the serial multipliers in a given row of FIG. 8 can be scaled by aconstant in order to reduce gate count and power consumption. Suchscaling does not affect the ELP roots. Additionally or alternatively,parallel multipliers 158 and/or 162 can be scaled by a constant, so thatregisters 166 begin the m-cycle sequence with a certain offset. Again,this scaling does not affect the ELP roots.

In some embodiments, each row of serial multipliers 170 in FIG. 8 can besplit into two halves, and each half operated to evaluate a separatefield element. Such a configuration is useful when the code wordcontains a small number of errors, such that the actual rank of the ELPis T/2 or less. In such a situation, unit 204 can evaluate 2d fieldelements concurrently, two field elements per row, thus reducing powerconsumption and latency. In order to support this sort of functionality,XOR unit 212 should calculate the XOR of the interim results of eachhalf row separately.

Reducing Power Consumption of Root Search Based on Actual ELP Rank

In many practical cases, the actual number of errors j in a given codeword is smaller than the maximum number of correctable errors T. Inother words, the rank of the ELP may be lower than T. In someembodiments, the ELP root search unit modifies its operation based onthe actual ELP rank, so as to reduce power consumption.

FIG. 9 is a block diagram that schematically illustrates an ELP rootcalculation unit 216, in accordance with an embodiment of the presentinvention. The ELP coefficients are stored in T registers 220. Unit 216applies the ELP to the different Galois field elements using GFmultipliers 224 and registers 228. The outputs of registers 228 arecombined by a GF adder 232, whose output determines whether thecurrently-evaluated is an ELP root or not. For example, unit 216 mayscan the Galois field elements using the Chien search process, citedabove. The present example refers to m=16, although any other suitablevalue of m can also be used.

Unit 216 further comprises a control unit 236, which controls thedifferent components of unit 216. In particular, control unit 236provides clock signals to the different multipliers and registers ofunit 216. The configuration of registers and multipliers in FIG. 9 is anexample configuration. The techniques described below can be applied inany other suitable ELP root search configuration having multipliersregisters and other logic components, such as the configurations ofFIGS. 6 and 8 above.

The power consumption of unit 216 depends primarily on the powerconsumption of registers 228 and multipliers 224. For example, thedifferent bits of registers 228 flip their values during operation ofunit 216, and this toggling has a considerable impact on powerconsumption. In particular, when the registers are implemented usingComplementary Metal Oxide Semiconductor (CMOS) technology, they consumepower mainly during value transitions.

When the actual rank j of the ELP is smaller than T, the root search canbe carried out using only a subset of the T registers and multipliers.In some embodiments, control unit 236 supplies clock signals selectivelyto the different logic components (e.g., multipliers 224 and registers228), such that a clock signal is provided only to j of the multipliersand registers. The remaining T−j multipliers and registers do notreceive clock signals, and their power consumption in minimized. (Atypical CMOS flip-flop circuit consumes approximately 30% of its powerirrespective of whether its value flips or not, as long as it receives aclock signal.)

FIG. 10 is a flow chart that schematically illustrates a method forlocating ELP roots, in accordance with an embodiment of the presentinvention. The description that follows focuses on the identification ofELP roots for a given code word. The method of FIG. 10 begins withcontrol unit 236 in root search unit 216 accepting an indication of theactual ELP rank j, at a rank input step 240. The rank can be accepted,for example, from an ELP computation unit that computes the ELPcoefficients (e.g., unit 98 in FIG. 3 above).

Control unit 236 checks whether the actual rank is smaller than themaximum allowable rank T, at a rank checking step 244. If j<T, thecontrol unit disables the clock signals to the T−j multipliers andregisters corresponding to the most significant ELP coefficients, at aselective clock disabling step 248.

In some embodiments, the control unit can also modify the rate of theclock signal based on the actual ELP rank (i.e., based on the number ofactive multipliers and registers), at a clock rate setting step 252.Using this technique, the control unit can limit the peak powerconsumption of unit 216. For example, when j is high and therefore alarge number of multipliers and registers are active, control unit 236may set a relatively low clock rate in order to limit the peak powerconsumption. When j is low, the control unit may set a higher clockrate, since only a relatively small number of multipliers and registersare active. Because of the different clock rates used, code words havinglarge numbers of errors will typically be decoded more slowly than codewords having small numbers of errors. Since code words having largenumbers of errors are relatively rare, this degradation is decoding timeis typically tolerable. In some embodiments, control unit 236 definestwo or more clock rates, which correspond to different ranges of the ELPrank j. For example, the control unit may apply a certain fast clockrate when j<j1, divide the fast clock by two for j1<j<j2, divide thefast clock by three when j2<j<j3, and so on (j1<j2<j3).

Unit 216 locates the j ELP roots using the active multipliers andregisters, at a root location step 256. (If rank checking step 244concludes that j=T, the method jumps directly to step 256, withoutdisabling clock signals or modifying the clock rate.)

FIG. 11 is a flow chart that schematically illustrates another methodfor locating ELP roots, in accordance with an embodiment of the presentinvention. In this method, when an ELP root is identified, control unit236 divides the ELP by a factor that depends on the identified root, soas to produce a lower rank ELP. The search process continues using thelower rank ELP. When using the method of FIG. 11, the power consumptionof unit 216 decreases over time, since the ELP rank decreases with eachadditional root found. The decrease in power consumption is particularlysignificant when this method is used in combination with the techniquesof FIG. 10 above.

The method of FIG. 11 begins with unit 216 searching for the roots of acertain ELP(x), at a root search step 260. Control unit 236 checkswhether a root has been found, at a root checking step 264. When a rootβ is found, control unit 236 divides ELP(x) by (x−β), at a division step268. Any suitable polynomial division method can be used for thispurpose. Example methods are described by Blahut in “Theory and Practiceof Error Control Codes,” Addison-Wesley, 1983, chapter 6, sections6.1-6.3, pages 130-140, which is incorporated herein by reference. Sinceβ is a root of ELP(x), the division has a remainder of zero. Thepolynomial division lowers the rank of the ELP by 1, without affectingthe ELP roots that have not yet been found.

Control unit 236 loads registers 220 with the coefficients of the new,lower-rank ELP. The method then loops back to step 260 above, and unit216 continue to search for additional ELP roots of the lower-rank ELP.The process continues until all ELP roots are found.

In some embodiments, unit 236 may end the progressive polynomialdivision when the ELP rank reaches a certain value (e.g., 1, 2 or 3).From that point, the control unit finds the remaining ELP roots usingother means, such as using algebraic equation solving methods. Thistechnique further reduces the search time.

When ECC decoder 90 operates on a set of input code words, some of thesecode words may contain more than T errors per code word. The decoding ofsuch code words will typically fail, since the number of errors per codeword exceeds the correction capability of the code. Typically,attempting to decode a code word having more than T errors will producea rank T ELP, but the root search process will fail.

In some embodiments, decoder 90 defines a certain upper bound T₀<T, andattempts to decode only code words whose ELP rank is no more than T₀. T₀may be set, for example, to T−1, T−2 or to any other suitable value.This technique eliminates attempts to decode non-decodable code words(which typically translate to rank T ELPs), and the unnecessary powerconsumption associated with these attempts. On the other hand, codewords whose number of errors is between T₀ and T will also not bedecoded. In other words, the effective correction capability of the codeis reduced from T to T₀. In most practical cases, however, such codewords are relatively rare and the resulting performance degradation istolerable and is well worth the reduction in power consumption.

Although the embodiments described herein mainly address decoding of BCHand RS codes in communication and storage applications, the methods andsystems described herein can also be used in any other suitableapplication.

It will thus be appreciated that the embodiments described above arecited by way of example, and that the present invention is not limitedto what has been particularly shown and described hereinabove. Rather,the scope of the present invention includes both combinations andsub-combinations of the various features described hereinabove, as wellas variations and modifications thereof which would occur to personsskilled in the art upon reading the foregoing description and which arenot disclosed in the prior art.

What is claimed is:
 1. An memory controller, comprising: an ErrorCorrection Code (ECC) unit configured to: receive data from a memory,wherein the data is encoded with an ECC, and includes at least one codeword; calculate a syndrome for the at least one code word; calculate anError Locator Polynomial (ELP) dependent upon the calculated syndrome;and modify a frequency of a clock signal dependent upon an actual numberof errors in the at least one code word; and a processing unit coupledto the ECC unit, wherein the processing unit is configured to: receivethe clock signal; and determine roots of the ELP.
 2. The apparatus ofclaim 1, wherein to calculate the syndrome for the at least one codeword, the ECC unit is further configured to multiply the at least onecode word by a parity matrix.
 3. The apparatus of claim 1, wherein tomodify the frequency of the clock signal, the ECC unit is furtherconfigured to decrease the frequency of the clock signal responsive to adetermination that the actual number of errors in the at least one codeword is greater than a predetermined threshold value.
 4. The apparatusof claim 1, wherein to modify the frequency of the clock signal, the ECCunit is further configured to increase the frequency of the clock signalresponsive to a determination that the actual number of errors in the atleast one code word is less than a predetermined threshold value.
 5. Theapparatus of claim 1, wherein to modify the frequency of the clocksignal, the ECC unit is further configured to compare the actual numberof errors in the at least one code word to a maximum number ofcorrectable errors.
 6. The apparatus of claim 1, wherein to calculatethe syndrome, the ECC unit is further configured to apply vectoroperations in a vector space to data bits included in the at least onecode word.
 7. The apparatus of claim 6, wherein syndrome is defined overa field, and wherein the field includes a primitive element.
 8. A methodfor operating a memory controller, the method comprising: receiving datafrom a memory, wherein the data is encoded with an Error Correction Code(ECC), and wherein the received data includes at least one code word;calculating a syndrome for the at least one code word; calculating anError Locator Polynomial (ELP) dependent upon the calculating syndrome;modifying a clock rate of a processing unit dependent upon an actualnumber of errors in the at least one code word; and determining, by theprocessing unit, roots of the ELP dependent upon the clock signal. 9.The method of claim 8, wherein calculating the syndrome for the at leastone code word comprising multiplying the code word by a parity matrix.10. The method of claim 8, wherein modifying the clock rate of theprocessing unit comprised decreasing the clock rate of the processingunit responsive to a determination that the actual number of errors inthe at least one code word is greater than a predetermined thresholdvalue.
 11. The method of claim 8, wherein modifying the clock rate ofthe processing unit comprised increasing the clock rate of theprocessing unit responsive to a determination that the actual number oferrors in the at least one code word is less than a predeterminedthreshold value.
 12. The method of claim 8, wherein modifying the clockrate of the processing unit comprises comparing the actual number oferrors in the at least one code word to a maximum number of correctableerrors.
 13. The method of claim 8, wherein calculating the syndromecomprises applying vector operations in a vector space to data bitsincluded in the at least one code word.
 14. The method of claim 13,wherein the syndrome is defined over a field, and wherein the fieldincludes a primitive element.
 15. A data storage apparatus, comprising:a memory unit; and a controller coupled to the memory unit, wherein thecontroller is configured to: receive data from the memory, wherein thedata is encoded with an ECC, and includes at least one code word;calculate a syndrome for the at least one code word; calculate an ErrorLocator Polynomial (ELP) dependent upon the calculated syndrome; andmodify a frequency of a clock signal dependent upon an actual number oferrors in the at least one code word; and a processor configured to:receive the clock signal; and determine roots of the ELP.
 16. The datastorage apparatus of claim 15, wherein to calculate the syndrome for theat least one code word, the controller is further configured to multiplythe at least one code word by a parity matrix.
 17. The data storageapparatus of claim 15, wherein to modify the frequency of the clocksignal, the controller is further configured to decrease the frequencyof the clock signal responsive to a determination that the actual numberof errors in the at least one code word is greater than a predeterminedthreshold value.
 18. The data storage apparatus of claim 15, wherein tomodify the frequency of the clock signal, the controller is furtherconfigured to increase the frequency of the clock signal responsive to adetermination that the actual number of errors in the at least one codeword is less than a predetermined threshold value.
 19. The data storageapparatus of claim 15, wherein to modify the frequency of the clocksignal, the controller is further configured to compare the actualnumber of errors in the at least one code word to a maximum number ofcorrectable errors.
 20. The data storage apparatus of claim 15, whereinto calculate the syndrome, the controller is further configured to applyvector operations in a vector space to data bits included in the atleast one code word.